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The curvature and torsion: how to distinguish the shape of a curve

Curvature and Torsion

Definition:
Let \(f:\, I\rightarrow\mathbb{R}^{3}\) be a \(C^{2}\) regular curve
parameterized by arc length \(s\in I\). The curvature of \(f\)
is the function \(k\) given by \[k(s)=\left|f'(s)\right|\]

Definition:
Let \(f:\, I\rightarrow\mathbb{R}^{3}\) be a \(C^{2}\) regular curve parameterized by arc length
\(s\in I\). The torsion of
\(f\) is the function \(\tau\) defined on the points where
\(f''(s)\neq0\) (that is, where the curvature is strictly positive),
such that
\[B'(s)=-\tau(s)N(s).\]